Nonlinear internal resonance based micromechanical resonators

ABSTRACT

A micromechanical resonator having a structure defining a first mode and a second mode and permitting non-linear internal resonance between the first and second modes. The resonator may further include a first component embodying a first mode and a second component embodying a second mode such that the first and second modes are substantially completely non-linearly coupled with each other while the second component vibrates at a frequency approximately twice the at least one first natural resonance frequency.

CROSS-REFERENCE TO RELATED APPLICATION

This patent application claims the benefit under 35 U.S.C. § 119(e) ofU.S. provisional patent application Ser. No. 60/704,291, filed Aug. 1,2005 and entitled NONLINEAR INTERNAL RESONANCE BASED MICROMECHANICALRESONATORS, the entire contents of which are incorporated herein byreference.

BACKGROUND

1. Field of the Invention

The invention relates generally to a micromechanical resonator. Morespecifically, the invention relates to a micromechanical resonatorhaving nonlinear 1:2 internal resonance between any two linear modes ofthe mechanical resonator.

2. Related Technology

Micromechanical resonators constitute a key component of manymicroelectromechanical systems (MEMS) devices such as accelerometers,scanning force and atomic force microscopes (AFM), pressure andtemperature sensors, and microvibromotors for controlled movements atsmall scale. Nonlinearities play important role in the dynamics ofmicroresonators. For example, microstructures of microresonatorstypically experience geometric and inertial nonlinearities, which arecaused by the structure of the microresonators, and actuation mechanismnonlinearities, which are caused by the actuation mechanism of themicroresonators. As a more specific example, in the AFM cantileverprobes resonant dynamics, van der Waals interactions are shown to leadto a softening nonlinear response while the short range repulsive forceslead to an overall hardening response. An inaccurate representation ofnonlinearities can lead to an erroneous prediction of the frequencyresponse and potentially failure of design based on the erroneoussimulations.

Radio frequency (RF) filters are commonly used for various applications,such as wireless applications and hand-held communicator devices. RFfilters typically include a filter component that receives input signalsand filters out all or substantially all signals having a frequencyother than a desired frequency. Additionally, RF filters often include amixing component that adjusts the output frequency after the filteringoperation has occurred. RF filters are conventionally made usingcrystals, which are relatively bulky and which consume a relativelylarge amount of power.

It is therefore desirable to design a microresonator that permits theuser to utilize nonlinearities in improving the microresonatorperformance. It is also therefore desirable to provide a microresonatorbased RF filter having a reduced size and reduced power consumptionwhile being capable of simultaneously filtering and mixing incomingsignals.

SUMMARY

In overcoming the limitations and drawbacks of the prior art, thepresent invention provides a micromechanical resonator having astructure defining a first (lower) mode and a second (higher) mode andpermitting non-linear 1:2 internal resonance between the two designateddistinct modes.

In another aspect of the present invention, a micromechanical resonatoris provided, including a structural configuration having a firstcomponent embodying the first mode and a second component embodying thesecond mode. The first and second modes are substantially linearlydecoupled from each other while the second mode vibrates at a frequencytwice the natural frequency of the first mode.

In yet another aspect of the present invention, a micromechanicalresonator is provided, including a structure having a first componentdefining a first mode and a second component defining a second mode, andan actuator configured to resonantly excite the second component at asecond natural frequency. The second component is positioned withrespect to the first component such that resonant excitation of thesecond component at the second frequency induces resonant excitation ofthe first component at a first natural frequency.

Further objects, features and advantages of this invention will becomereadily apparent to persons skilled in the art after a review of thefollowing description, with reference to the drawings and claims thatare appended to and form a part of this specification.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 shows a plan view of a micromechanical resonator embodying theprinciples of the present invention;

FIG. 2 is a schematic representation of the structure shown in FIG. 1,showing deflecting components of the structure with respect to acoordinate system;

FIG. 3 is a graphical representation of natural frequencies of themicromechanical resonator shown in FIG. 1 as a function of the lengthratio of the beam segments for a particular set of values.

FIG. 4(a) is a schematic representation of the structure shown in FIG. 1where the first mode is activated;

FIG. 4(b) is a schematic representation similar to FIG. 4(a), where thesecond mode is activated;

FIG. 5(a) is a graphical representation of the displacement of the tipof the vertical beam of the T-beam structure shown in FIG. 1 as afunction of the excitation frequency, where the first mode is in theprimary resonance;

FIG. 5(b) is a graphical representation of the vertical displacement ofthe junction of the three beams of the T-beam structure shown in FIG. 1as a function of the excitation frequency, where the first (lowerfrequency) mode is directly excited by external means and is in primaryresonance;

FIG. 6(a) is a graphical representation similar to FIG. 5(a), where thesecond (higher frequency) mode is directly excited by external actuationand is in the primary resonance;

FIG. 6(b) is a graphical representation similar to FIG. 5(b), of thevertical displacement of the junction of the three beams of the T-beamstructure when the second (higher frequency) mode is in primaryresonance;

FIG. 7(a) is a graphical representation of the displacement of the uppertip of the vertical beam shown in FIG. 1 as a function of excitationfrequency, where the first (lower frequency) mode is directly excited inprimary resonance;

FIG. 7(b) is a graphical representation of the displacement similar toFIG. 7(a), when the second (higher frequency) mode is directly excitedin primary resonance;

FIG. 8 is a plan view of a second embodiment of a micromechanicalresonator embodying the principles of the present invention; and

FIG. 9 is a plan view of a third embodiment of a micromechanicalresonator embodying the principles of the present invention.

DETAILED DESCRIPTION

Referring now to the drawings, FIG. 1 is a schematic of a microresonator10 according to a first embodiment of the present invention. Themicrostructure 10 generally includes a structure 11 defined by a firstcomponent 12 and a second component 14, a base 15 (or substrate)supporting the structure 11, a first mass 16 and a second mass 18coupled with the structure 11, a first electrode 24 positioned adjacentto the first component 12, and a second electrode 20 positioned adjacentto the second component 12.

The second component 14 of the structure 11 includes a first horizontalbeam 26 and a second horizontal beam 28, each having a first endconnected to the base 15 and a second end connected to the first mass16. Each of the horizontal beams 26, 28 may be connected to the base 15that does not permit displacement or pivoting movement of the end of thebeam 26, 28 at the base 15. Similarly, each horizontal beam 26, 28 mayalso be connected to the first mass 16 by a fixed connection. Thehorizontal beams 26, 28 shown in FIG. 1 both have equal or substantiallyequal specifications, such as length, width, thickness, diameter, andtype of material, so the respective horizontal beams 26, 28 performsubstantially identically when an external force is applied equallythereto.

The first component 12 of the structure 11 includes a vertical beam 30coupled with the second component 14 of the structure 11 by the firstmass 16. More specifically, one end of the vertical beam 30 may beconnected to the midpoint of the first mass 16 so that the vertical beam30 is centered along the length of the second component 14 and so that avertical axis 32 of the first component 12 is generally perpendicular toa horizontal axis 34 of the second component 14. The other (free) end ofthe vertical beam 30 is connected to the second mass 18. The verticalbeam 30 may be fixed to the first mass 16 by a fixed connection.

The second electrode 20 is an electrostatic electrode positionedadjacent to, and centered along the length of, the horizontal beams 26,28. The second electrode 20 is connected to an electrical power supplythat provides a bias voltage V_(b) and a harmonically fluctuatingvoltage V_(ac) sufficient, depending upon the quality factor determinedby operating conditions, for selectively inducing resonance in thesecond component 14. For the structure with specific dimensions givenbelow, and for a 1 um gap between the electrode 20 and the structure 26,28 with quality factor Q=5000, Vproductthreshold=0.05Vˆ2, that isV_(b)=5V, and V_(AC)=0.01V.

Upon activation of the second electrode 20, an electro-static field isgenerated, thereby causing the second component 14 to deflect and definea second mode 40 (FIGS. 2 and 4(b)). As is known in the art, a mode isdefined as the movement profile of an elastic body in which the bodymoves at a natural frequency. For example, the second (higher frequency)mode 40 defines a generally sinusoidal (in time) vertical deflection ofthe respective horizontal beams 26, 28 and displacement of the firstmass 16 in response to the external force from the alternating currentof the power supply. In this second mode, the vertical component 12 andthe mass 18 also move axially along the axis 32 without flexing ordeforming, as shown in FIG. 2. More specifically, the power supplycauses the first mass 16 to move, and the horizontal beams 26, 28 todeflect, towards and away from the second electrode 20. Although for thespecific design configuration of the present invention shown in FIG. 1,the second component 14 (see FIG. 1) embodies a single mode when thesecond electrode 20 is activated, the present invention may also beutilized in a microresonator in which a single physical componentembodies two or more modes that are in 1:2 internal resonance.

When the input frequency of the power supply is not equal orapproximately equal to a natural frequency of the higher mode ofvibration (embodied in deflection of the second component 14), theamplitude of deflection of the second component 14 will be relativelylow. Conversely, when the input frequency is equal or approximatelyequal to a natural resonance frequency of the second component 14, thenthe second component 14 will undergo resonance having a relatively highamplitude.

As a result of the deflection of the second component 14 in higherfrequency mode of the microresonator, the first component 12 is likewisedisplaced along the vertical axis 32. More specifically, because each ofthe two horizontal beams 26, 28 deflects by an equal distance, thevertical beam 30 moves generally vertically along the vertical axis 32.

Similarly to the second electrode 20, the first electrode 24 isconfigured to generate an electrostatic field. This electrostatic fieldis able to detect disruptions in the electrostatic field caused byhorizontal movement of the vertical beam 30. Thus, the first electrode24 is able to detect deflection of the vertical beam 30 transverse tothe vertical axis 32 (parallel to the horizontal axis 34). In analternative configuration, the first electrode 24 operates to exciteresonance in the lower frequency mode of the microresonator, whichincludes deflection of the first beam 12 while the second electrode 20detects movement of the second beam 14. In another alternative design, apair of electrodes is positioned adjacent to the second beam 30, one forexciting transverse (parallel to axis 34) movement of the second beam 30and one for measuring the deflection of the second beam 30.

As mentioned above, when the second component 14 is not resonating, thefirst component 12 will undergo little or no deflection parallel to thehorizontal axis 34. More specifically, when the second component 14 isnot resonating, the first component 12 will remain generally straightand perpendicular to the first mass 16 so that the horizontal distancebetween the first beam 30 and the first electrode 24 remains relativelyconstant. As a result of the constant horizontal distance between thefirst beam 30 and the first electrode 24, the first electrode 24 willdetect little or no change to the electrostatic field. Therefore, whenthe microresonator, and hence the second component 14, vibrates at afrequency other than a natural frequency, the second component 12 willnot deflect horizontally. In other words, the higher frequency and thelower frequency natural modes, with specific examples shown in the formof modes 40, 42 in FIGS. 2, 4(a), and 4(b), are coupled with each otherwhile the second component 14 vibrates at a frequency other than thehigher (second) natural resonance frequency. This characteristic makesthe structure 11 particularly useful as a filter because the firstelectrode 24 will detect little or no change in the electrostatic fieldadjacent to the first component 12 when the second component 14 isvibrating at a frequency other than the higher (second) naturalfrequency. Therefore, in this operational configuration, themicroresonator 12 can be used to filter out all frequencies that are notvery close to the higher natural frequencies of the microresonator,vibrating in second mode depicted in FIG. 4(b). In this higher mode,only the second component 14 flexes, as shown in FIG. 4(b).

Conversely, the first and second modes are substantially linearlydecoupled from each other while the second mode vibrates at a frequencyapproximately twice the natural frequency of the first mode. In otherwords, due to 1:2 internal resonance which is discussed further below,when the microresonator is resonating in the higher frequency mode atthe natural frequency that is approximately 2 times larger than thenatural frequency of a lower frequency mode of the microresonator, thefirst component 12 will undergo horizontal deflection due to resonanceof the lower frequency mode, as signified by the mode 42 in FIG. 4(a).More specifically, the second (higher) mode 40 (as embodied by thesecond component 14 and the first component 12) excites the first(lower) mode 42 (as embodied by the first component 12 and the secondcomponent 14) and causes the first component 12 to resonate along thehorizontal axis in addition to vertical displacement along the verticalaxis 32 caused by the second mode 40. As the first component 12 isdeflected parallel to the horizontal axis 34, the first electrode 24 isable to detect deflection of the first component 12, thereby permittingthe microresonator 10 to serve as a sensing device, or a signalprocessing filtering device.

The lower and higher frequency modes (the first and second modes 42, 40)are coupled with each other through a phenomenon known as nonlinearmodal interaction. Non-linear modal interaction is defined as thephenomenon where one mode (in this case, the second mode 40) excitesanother mode (in this case, the first mode 42) through non-linearinteractions within the structure. The structure 11 utilizes modalinteractions in the two modes that arise due to inertial quadraticnonlinearities. Non-linear modal interaction in structure 11 occurs whenthe respective modes have natural frequencies having a ratio of 1:2. Forexample, if the second mode 40 (FIG. 4(b)) has a natural frequency of(f) and the first mode 42 (FIG. 4(a)) has a natural frequency of (f/2),then resonant excitation of the microresonator in its second mode 40 atfrequency (f) will induce resonance of the first mode at a frequency of(f/2). Experimental results show that non-linear modal interaction ismost effective when the ratio of natural frequencies is 1:2±0.5% so longas the Q factor is sufficiently high, as will be discussed in moredetail below.

In order for 1:2 resonance to occur between the respective modes, thefirst and second components 12, 14 must have a particular length ratioand/or weight ratio. More specifically, the first and second components12, 14 and the masses 16, 18 must have a suitable combination ofstructural characteristics for 1:2 resonance to occur. As a firstexample, the first mass 16 has a weight equal to the weight of the firstcomponent 12, the second mass 18 has a weight equal to the combinedweight of the horizontal beams 26 and 28), and the first component 12has a length approximately equal to 108% (±2%) of one of the horizontalbeams 26, 28. In an alternative design, where no additional masses(masses 16 and 18) are present and the first and second components 12,14 each have the same thickness and are made of the same material, thefirst component 12 has a length approximately equal to 133% (±2%) of oneof the horizontal beams 26, 28.

The above scenario describes non-linear modal interaction when thehigher (second) frequency mode 40 (signified by flexing of the component14) is excited by an external force (the second electrode 20). However,non-linear modal interaction may also occur when the lower (first)frequency mode 42 (signified by flexing of both the components 12 and14) is resonantly excited by an external force (such as the firstelectrode 24), thereby inducing response of the second mode 40. Moreparticularly, if the first electrode 24 excites the first (lowerfrequency) mode 42 at a natural frequency that is ½ as large as anatural frequency of the second (higher frequency) mode 40, then thesecond mode 40 will be induced by the first mode 42.

In the above-described scenarios, non-linear modal interaction occursbetween the first and second modes. In a real working model, due tovariables such as damping or an imprecise 1:2 frequency ratio, theexternal force that excites one of the modes may have to have athreshold amplitude in order for non-linear modal interaction to occur.For example, structural damping inherent to the structure 11 or airdamping caused by components moving through the air may reduce theeffectiveness of the non-linear modal interaction, thereby reducing theoverall Quality of the system. Similarly, if the 1:2 ratio between thenatural frequencies of the two modes is not within an acceptable errorrange (such as ±0.5%) then the effectiveness of the non-linear modalinteraction may also be reduced. However, these system imperfections maybe overcome by increasing the amplitude of the input voltage to theelectrode.

Referring to FIGS. 1 and 2, mathematical models representing theabove-described micromechanical resonator will now be discussed in moredetail. The two horizontal beams 26, 28 have lengths denoted by L1, andL2 and the vertical beam 30 has a length denoted by L3. The first andsecond masses 16, 18 are denoted by M_(c) and M_(t) respectively.Because the microresonator works on the principle of 1:2 internalresonance, the linear analysis for this T-beam structure is performed todetermine the design conditions under which any two modes of themicroresonator, and more specifically the first and second modes, aretuned for 1:2 modal interaction. In the present analysis, the rotaryinertia terms for the beams are neglected as we are interested in onlythe lower modes (first two modes) of the beam structure. The motion isassumed to be in the horizontal plane (no gravity). The rotary inertiaof the rigid masses are taken into account without including thefiniteness (to simplify analysis) of these rigid masses. The secondelectrode 20 is assumed to span the bottom beam partially and, l₁ and l₂denote the span of the second electrode 20 over the horizontal beams 26,28. The second electrode 20 is located at a distance d from thehorizontal beams 26 and 28. The sensor electrode 24 has a length denotedby l₃ and is located at a distance d₃ from the vertical beam 30. Asmentioned above, these electrodes can be used for sensing the beamresponse or actuating the resonator depending upon the mode (shape) tobe excited.

The coordinate systems and displacements of beam segments are shown inFIG. 2. Axial and transverse displacements of a beam element in threebeams 26, 28, 30 are denoted by u_(i) and v_(i), where i refers to thebeam in consideration. The first horizontal beam 26 is beam 1, thesecond horizontal beam 28 is beam 2, and the vertical beam is beam 3. ALagrangian description is used in modeling this T-beam structure and asa result these displacements are functions of undeformed arc lengths_(i). The displacements for the horizontal beams 26, 28 are withrespect to the stationary substrate 15 and the displacements of thevertical beam 30 are measured with respect to the coordinate systemlocated at the junction of the three beam segments 26, 28, 30. Therotation of a beam element is denoted by Ψ_(i)(s_(i),t). The sheardeformation and warping are assumed to be negligible and thus, therotations of elements are related to the beam displacements as follows:$\begin{matrix}{{\sin\quad\psi_{i}} = \frac{\partial v_{i}}{\partial s_{i}}} & (1)\end{matrix}$

The kinetic energy T and potential energy V (including the electrostaticpotential) of the system are given by: $\begin{matrix}{\left. \begin{matrix}{T = {\left( {\sum\limits_{i = 1}^{2}{\int_{0}^{L_{i}}{\frac{1}{2}{m_{i}\left( {{\overset{.}{u}}_{i}^{2} + {\overset{.}{v}}_{i}^{2}} \right)}{\mathbb{d}s_{i}}}}} \right) +}} \\{{\int_{0}^{L_{3}}{\frac{1}{2}{m_{3}\left( {\overset{.}{u}}_{1} \middle| {}_{s_{1} = L_{1}}{- {\overset{.}{v}}_{3}} \right)}^{2}{\mathbb{d}s_{3}}}} +} \\{{\int_{0}^{L_{3}}{\frac{1}{2}{m_{3}\left( {\overset{.}{v}}_{1} \middle| {}_{s_{1} = L_{1}}{+ {\overset{.}{u}}_{3}} \right)}^{2}{\mathbb{d}s_{3}}}} +} \\{{\frac{1}{2}{M_{c}\left( \left. {\overset{.}{u}}_{1}^{2} \middle| {}_{s_{1} = L_{1}}{+ {\overset{.}{v}}_{1}^{2}} \right|_{s_{1} + L_{1}} \right)}} +} \\{\left. {\frac{1}{2}J_{c}{\overset{.}{\psi}}_{1}^{2\quad}} \middle| {}_{s_{1} = L_{1}} + \right.} \\{{\frac{1}{2}{M_{t}\left( \left. {\overset{.}{u}}_{1} \middle| {}_{s_{1} = L_{1}}{- {\overset{.}{v}}_{3}} \right|_{s_{3} = L_{3}} \right)}^{2}} +} \\{{\frac{1}{2}{M_{t}\left( \left. {\overset{.}{v}}_{1} \middle| {}_{s_{1} = L_{1}}{+ {\overset{.}{u}}_{3}} \right|_{s_{3} = L_{3}} \right)}^{2}} +} \\{\left. {\frac{1}{2}J_{t}{\overset{.}{\psi}}_{3}^{2}} \right|_{s_{3} = L_{3}}}\end{matrix} \right\}{and}} & (2) \\\left. \begin{matrix}{V = {{\sum\limits_{i = 1}^{3}{\int_{0}^{L_{i}}{\frac{1}{2}({EI})_{i}\left( \frac{\partial\psi_{i}}{\partial s_{i}} \right)^{2}{\mathbb{d}s_{i}}}}} +}} \\{{\sum\limits_{i = 1}^{2}{\int_{0}^{L_{i}}{\frac{1}{2}({EA})_{i}e_{i\quad 0}^{2}{\mathbb{d}s_{i}}}}} -} \\{{\left( {\int_{L_{3} - l_{3}}^{L_{3}}{\frac{1}{2}\frac{ɛ_{0}ɛ_{r}b_{3}}{d + v_{3}}\left( {V_{b} + {V\quad\cos\quad\left( {\Omega\quad t} \right)}} \right)^{2}{\mathbb{d}s_{3}}}} \right)M_{1}} -} \\{\left( {\sum\limits_{i = 1}^{2}{\int_{L_{i} - l_{i}}^{L_{i}}{\frac{1}{2}\frac{ɛ_{0}ɛ_{r}b_{i}}{d + v_{i}}\left( {V_{b} + {V\quad\cos\quad\left( {\Omega\quad t} \right)}} \right)^{2}{\mathbb{d}s_{i}}}}} \right)M_{2}}\end{matrix} \right\} & (3)\end{matrix}$where a dot denotes derivative with respect to time. The variables m_(i)and (EI)_(i) denote the mass per unit length and flexural rigidity,respectively, for the ith beam. The rotary inertia of the masses M_(c)and M_(t) are J_(c) and J_(t) respectively. The variable b_(i) denotesthe width of the ith beam. When the second electrode 20 is used foractuation, the variables M₁ and M₂ in expression (3) take on the valuesM₂=1 and M₁=0. When the first electrode 24 is used for actuation, thenM₂=0 and M₁=1. The strain along neutral axis in ith beam is denoted bye_(i0). The strain e_(i0) can also be expressed in terms of axial andtransverse displacements as follows: $\begin{matrix}{e_{i\quad 0} = {\sqrt{\left( {1 + \frac{\partial u_{i}}{\partial s_{i}}} \right)^{2} + \left( \frac{\partial v_{i}}{\partial s_{i}} \right)^{2}} - 1}} & (4)\end{matrix}$The inextensibility assumption for vertical beam 30 results in theconstraint e₃₀=0. In equation (3), the parameters ε₀ and ε_(r) in thethird term defining the electrostatic potential are permittivity ofspace (8.8504×10⁻¹² F/m) and the relative permittivity of dielectricbetween the gap (ε_(r)=1 for air gap) respectively. The voltage appliedbetween the second electrode 20 and the horizontal beams 26, 28 has a DCvoltage part denoted by V_(b) and an AC part with frequency Ω andvoltage amplitude V.

The augmented Lagrangian L accounting for the constraints is then asfollows: $\begin{matrix}\left. \begin{matrix}{L_{avg} = {T - V +}} \\{\frac{1}{2}{\int_{0}^{L_{3}}{{\lambda_{1}\left( {\left( {1 + \frac{\partial u_{3}}{\partial s_{3}}} \right)^{2} + \left( \frac{\partial v_{3}}{\partial s_{3}} \right)^{2} - 1} \right)}{\mathbb{d}s_{3}}}}}\end{matrix} \right\} & (5)\end{matrix}$where λ₁ is the Lagrange multiplier imposing the inextensibilityconstraint.Linear Analysis

First, a linear analysis of the structure will be discussed byevaluating the structure with small, finite amplitude oscillations. Thetransverse displacements v_(i) are scaled by a small dimensionlessparameter ε. The axial displacements are assumed to be caused bytransverse displacements and are of O(ε²). This essentially means thataxial motion rigidity (EA)_(i) is much larger than flexural rigidity(EI). These scalings are used to order nonlinear terms and only up toquadratic nonlinearities are retained in the present equations. As itwill later turn out, the forcing will be scaled as O(ε²), and it will beseen that the effect of electrostatic actuation (including the non zeroequilibrium position of the beam due to DC voltage) on linear naturalfrequencies will be of higher order. When retaining terms only up to thequadratic nonlinearities in the system (terms up to O(ε³)), theelectrostatic actuation will only result in change in static equilibriumposition.

The linear equations of motion are obtained by introducing thesescalings in Lagrangian, Equation (5), then retaining terms up to theorder of O(ε²) and using Hamilton's principle. The non-dimensionalizedlinear equations of motion turn out to be as follows: $\begin{matrix}{{{{\overset{=}{v}}_{i} + {\frac{\alpha_{i}}{r_{i}v_{i}^{4}}\frac{\partial^{4}{\overset{\_}{v}}_{i}}{\partial{\overset{\_}{s}}_{i}^{4}}}} = 0},} & (6)\end{matrix}$where a dot now represents a derivative with respect to thenon-dimensional time τ. These non-dimensional parameters are defined asfollows: $\begin{matrix}\left. \begin{matrix}{{{\overset{\_}{v}}_{i} = \frac{v_{i}}{L}},\quad{{\overset{\_}{s}}_{i} = \frac{s_{i}}{L_{i}}},\quad{\alpha_{i} = \frac{({EI})_{i}}{EI}}} \\{{r_{i} = \frac{m_{i}}{M}},\quad{v_{i} = \frac{L_{i}}{L}},\quad{\tau = {\sqrt{\frac{EI}{{ML}^{4}}}{t.}}}}\end{matrix} \right\} & (7)\end{matrix}$In defining these non-dimensional parameters of the system, M is anominal mass per unit length, L is a nominal length, and EI is a nominalflexural rigidity. The arc length s_(i) of the ith beam isnondimensionalized using the length of the corresponding beam. Thus, theequations of motion are valid over the region O< s<1. Further, thetransverse displacements are measured from the static equilibriumposition of the beam which is changed due to electrostatic actuation.Thus, the formulation here assumes that the oscillations of the beam areabout non-zero equilibrium position of the beam; however the non-zeroequilibrium position effect on the natural frequencies are of higherorder. The electrostatic potential terms are non-dimensionalized usingthe following scalings: $\begin{matrix}\left. \begin{matrix}{{g = \frac{d}{L}},\quad{{\overset{\_}{l}}_{i} = \frac{l_{i}}{L_{i}}}} \\{F_{0} = {\left( {\frac{ɛ_{0}ɛ_{r}b_{1}}{g}\left( {V_{b}^{2} + \frac{V^{2}}{2}} \right)} \right)/\left( {{EI}/L} \right)}} \\{F_{1} = {\left( {\frac{ɛ_{0}ɛ_{r}b_{1}}{g}\left( {2V_{b}V} \right)} \right)/\left( {{EI}/L} \right)}} \\{F_{2} = {\left( {\frac{ɛ_{0}ɛ_{r}b_{1}}{g}\left( \frac{V^{2}}{2} \right)} \right)/\left( {{EI}/L} \right)}}\end{matrix} \right\} & (8)\end{matrix}$where g is the non-dimensional gap between the structure and thestationary electrode, I _(i) is the non-dimensional span of theelectrode over ith beam, F₀ relates to static force, F₁ and F₂ relate toharmonic forces with frequencies Ω and 2Ω respectively.

Ideal clamp assumption at the two ends of the horizontal beam constrainsthe slope and displacement to be zero at S ₁0 and S ₂=0. Also, thedisplacement of the upper beam is measured from the coordinate at thebeginning of the upper beam and as a result the displacement ν ₃=0 at S₃=0. Apart from these five boundary conditions, the rest of the boundaryconditions are as listed below: $\begin{matrix}{{\left. \frac{\partial{\overset{\_}{v}}_{1}}{\partial{\overset{\_}{s}}_{1}} \right|_{s_{1} = 1} = \left. {\frac{v_{2}}{v_{1}}\frac{\partial{\overset{\_}{v}}_{2}}{\partial{\overset{\_}{s}}_{2}}} \right|_{s_{2} = 1}},} & (9) \\{{\left. {\overset{\_}{v}}_{1} \right|_{{\overset{\_}{s}}_{1} = 1} = \left. {- {\overset{\_}{v}}_{2}} \right|_{{\overset{\_}{s}}_{2} = 1}},} & (10) \\{{\left. \frac{\partial{\overset{\_}{v}}_{1}}{\partial{\overset{\_}{s}}_{1}} \right|_{{\overset{\_}{s}}_{1} = 1} = \left. {\frac{v_{3}}{v_{1}}\frac{\partial{\overset{\_}{v}}_{3}}{\partial{\overset{\_}{s}}_{3}}} \right|_{{\overset{\_}{s}}_{3 = 0}}},} & (11) \\\left. \begin{matrix}{\left. {\frac{\alpha_{1}}{v_{1}^{3}}\frac{\partial^{2}{\overset{\_}{v}}_{3}}{\partial{\overset{\_}{s}}_{2}^{2}}} \middle| {}_{{\overset{\_}{s}}_{1} = 1}{{- \frac{\alpha_{2}}{v_{3}^{3}}}\quad\frac{\partial^{2}\quad{\overset{\_}{v}}_{1}}{\partial{\overset{\_}{s}}_{1}^{2}}} \right|_{{\overset{\_}{s}}_{1}\quad = \quad 1} =} \\\left. {\left( {{\left( {1 + R_{t}} \right)r_{3}v_{3}} + {R_{c}\left( {{r_{1}v_{1}} + {r_{2}v_{2}}} \right)}} \right){\overset{\overset{..}{\_}}{v}}_{1}} \right|_{s_{1} = 1}\end{matrix} \right\} & (12) \\\left. \begin{matrix}\left. {\frac{\alpha_{3}}{v_{3}^{2}}\frac{\partial^{2}{\overset{\_}{v}}_{3}}{\partial{\overset{\_}{s}}_{3}^{2}}} \middle| {}_{{\overset{\_}{s}}_{3} = 1}{{- \frac{\alpha_{1}}{v_{1}^{2}}}\quad\frac{\partial^{2}{\overset{\_}{v}}_{1}}{\partial{\overset{\_}{s}}_{1}^{2}}} \middle| {}_{{\overset{\_}{s}}_{1}\quad = \quad 1} - \right. \\{{\left. {\frac{\alpha_{2}}{v_{2}^{2}}\quad\frac{\partial^{2}{\overset{\_}{v}}_{2}}{\partial{\overset{\_}{s}}_{2}^{2}}} \right|_{{\overset{\_}{s}}_{2}\quad = \quad 1} = \left. {\frac{\gamma_{c}}{v_{1}}\frac{\partial^{3}{\overset{\_}{v}}_{1}}{{\partial\tau^{2}}{\partial{\overset{\_}{s}}_{1}}}} \right|_{{\overset{\_}{s}}_{1} = 1}},}\end{matrix} \right\} & (13) \\{{\left. {\frac{\alpha_{3}}{v_{3}}\frac{\partial^{2}{\overset{\_}{v}}_{3}}{\partial{\overset{\_}{s}}_{3}^{2}}} \right|_{{\overset{\_}{s}}_{3} = 1} = \left. {R_{t}r_{3}v_{3}{\overset{..}{\overset{\_}{v}}}_{3}} \right|_{s_{3} = 1}},} & (14) \\{{\left. {\frac{\alpha_{3}}{v_{3}}\frac{\partial^{2}{\overset{\_}{v}}_{3}}{\partial{\overset{\_}{s}}_{3}^{2}}} \right|_{{\overset{\_}{s}}_{3} = 1} = \left. {{- \gamma_{t}}\frac{\partial^{3}{\overset{\_}{v}}_{3}}{{\partial\tau^{2}}{\partial{\overset{\_}{s}}_{3}}}} \right|_{{\overset{\_}{s}}_{3} = 1}},} & (15)\end{matrix}$where non-dimensional parameters (R_(t),γ_(t)) and (R_(c),γ_(c)),related to the rigid masses M_(c) and M_(t) respectively, as defined by:$\begin{matrix}\left. \begin{matrix}{{\gamma_{t} = \frac{\quad J_{\quad t}}{\quad{ML}^{\quad 3}}},} & {{R_{t} = \frac{M_{t}}{m_{3}L_{3}}},} \\{{\gamma_{c} = \frac{J_{c}}{{ML}^{3}}},} & {R_{c} = {\frac{M_{c}}{\left( {{m_{1}L_{1}} + {m_{2}L_{2}}} \right)}.}}\end{matrix} \right\} & (16)\end{matrix}$

Boundary condition in equation (9) ensures that the slopes of the twohorizontal beams 26, 28 are equal at the junction of these beams 26, 28.Equation (10) constrains the bottom two beams to have the sametransverse displacement at the junction. The negative sign in thisequation appears as the coordinate system for the left and right bottombeams are different. Equation (11) constrains the vertical beam 30 to beperpendicular to the horizontal beams 26, 28 at the junction thereof.

The boundary conditions in equations (12) and (15) can be derived byeither doing the force and moment balance at the junction or byintroducing the geometric boundary conditions in equations (9)-(11) asconstraints in the Lagrangian using three more Lagrange multipliers andthen eliminating Lagrange multipliers to determine the boundaryconditions. The shear forces due to bending in the horizontal beams 26,28 support the inertial force due to the displacement of the first mass16, the second mass 18, and the vertical beam 30. This force balance atthe junction results in the boundary conditions (12) and similarly themoment balance at the junction gives boundary condition in equation(13). The boundary conditions in equations (14) and (15) correspond tothe force and moment balance at the tip of the vertical beam 30.

The linear mode shapes and natural frequencies are obtained by assumingthe solution to have the following form:ν _(i)( s _(i) ,t)=V _(i)( s _(t))F(t),  (17)where V_(i) is a spatial dependent function and F(t) is a harmonicfunction with frequency at ω. Substituting the assumed solution (17)into the governing equations (6), and separating space and time, we findthat the following solution satisfies the governing equations:$\begin{matrix}\left. \begin{matrix}{{V_{i}\left( {\overset{\_}{s}}_{i} \right)} = {{a_{i}\quad\cos\quad\beta_{i}{\overset{\_}{s}}_{i}} + {b_{i}\quad\sin\quad\beta_{i}{\overset{\_}{s}}_{i}} +}} \\{{{c_{i}\cosh\quad\beta_{i}{\overset{\_}{s}}_{i}} + {d_{i}\sinh\quad\beta_{i}{\overset{\_}{s}}_{i}}},}\end{matrix} \right\} & (18)\end{matrix}$where β_(i) is given by: $\begin{matrix}{\beta_{i}^{4} = {\frac{r_{i}v_{i}^{4}\omega^{2}}{\alpha_{i}}.}} & (19)\end{matrix}$The boundary conditions (at clamped ends and junction, and (9)-(15)) areused to now determine a characteristic matrix whose determinant is thecharacteristic equation. The roots of the characteristic equationdetermine natural frequencies (ω) and linear mode shapes aresubsequently obtained for these natural frequencies by using thecharacteristic matrix. Thus, the exact linear mode shapes are obtainedanalytically.

The purpose of introducing first and second masses 16, 18 in theanalysis is to make the model flexible enough to achieve desired designobjectives. More specifically, as discussed above, to achieve 1:2resonance without the additional masses 16, 18, the length ratios of thefirst and second components 12, 14 must be approximately 0.66 to 0.68.Once it is decided to keep the rigid masses or not, the linear analysispresented here can be used to identify conditions for which thestructure exhibits 1:2 internal resonance and to obtain mode shapes andnatural frequencies. The example used in this paper to illustrate theresults of the analysis (linear and nonlinear) does not include anyrigid mass; however, the formulation includes the rigid mass to keep theanalysis presented here relevant to designs requiring the use of rigidmasses. Another example discussed above for a 1:2 internal resonancebetween the two lowest modes of the microresonator is: the first mass 16has a weight equal to the weight of the first component 12, the secondmass 18 has a weight equal to the combined weight of the horizontalbeams 26 and 28), the first component 12 has a length approximatelyequal to 108% (±2%) of one of the horizontal beams 26, 28.

Now, consider the specific system with no rigid masses and all threebeams having the same mass per unit length and flexural rigidity. Also,we assume that the lengths of the horizontal beams are equal, L₁=L₂ orv₁=v₂. Thus, the only parameter that is not fixed is the ratio of thelength of vertical beam to the length of one of the bottom beams(v₃/v₁). Natural frequency is computed analytically for different valuesof the length ratio (v₃/v₁). We found that when the ratio of the lengthof upper beam to the length of one of the bottom beams is 1.3266, thesecond natural frequency is twice that of the first natural frequency.This provides us with an important design condition to have 1:2 internalresonance. This critical ratio is denoted by$\left( \frac{v_{3}}{v_{1}} \right)_{c},$with $\left( \frac{v_{3}}{v_{1}} \right)_{c} = {1.3266.}$

An ANSYS FEM model of T-beam structure is used to verify the abovenon-dimensional linear analysis. Beam elements are used in modeling thestructure in ANSYS. As for the linear analysis, electrostatic actuationeffects are considered of higher order, the FEM model also does notinclude electrostatic actuation. The dimensions of the structure withmaterial properties of polysilicon are as follows: $\begin{matrix}\left. \begin{matrix}{{L_{1} = {30\quad{µm}}},} & {{L_{3} = {30\frac{v_{3}}{v_{1}}\quad{µm}}},} \\{{E_{1} = {150 \times 10^{9}N\text{/}m^{2}}},} & {{b_{1} = {3\quad{µm}}},} \\{{I_{1} = {0.84375\quad({µm})^{4}}},} & {m_{1} = {0.010485{\frac{Kg}{({µm})}.}}}\end{matrix} \right\} & (20)\end{matrix}$Also, all the beams have the same cross-sectional area and are made ofthe same material. The first four natural frequencies for this systemfor different values of the length ratio ν₃/ν₁ are shown in FIG. 3. Theanalytically computed mode shapes for the critical length$\left( \frac{v_{3}}{v_{1}} \right)_{c} = 1.3266$are shown in FIG. 4. In the first (or the lower frequency) mode (mode42), the horizontal beam moves very little as compared to the verticalbeam. However, in the second mode (mode 40), the transverse displacementof the upper beam is zero. Due to the nature of the modes, note that theresponse in the second mode will not result in any deflection ofvertical beam, unless energy is transferred from second mode to thefirst mode through 1:2 internal resonance. The nonlinear responses ofthe structure with 1:2 internal resonance between the first two modesare presented in the next section for two cases of resonant excitations:(a) the resonant excitation of the first mode, and (b) the resonantexcitation of the second mode.Nonlinear Response under Resonant Excitation

The electrostatic actuation terms are scaled such that they are of O(ε²)as follows:F _(j)=ε² {circumflex over (F)} _(j)  (21)where j=0, 1, 2. This ordering ensures that the nonlinear terms in theLagrangian and resonant excitation are at the same order in theanalysis. The frequency of the first two modes of the structure arerelated to the actuation voltage frequency Ω as follows:Ω=R ₁ω₁(1+εσ₁),  (22)Ω=R ₂ω₂(1+εσ₂),  (23)where ω_(i) is the ith natural frequency of the structure, σ₁ and σ₂ arethe external detunings from perfect resonant excitation of either thefirst mode or the second mode. R₁ and R₂ determine the mode in primaryresonance. Specifically, for R₁=1 and R₂=½, the first mode is inresonance and for R₁=2 and R₂=1, the second mode is in resonance. Inusing these tuning criteria we have assumed that the DC voltage, V_(b)is not zero. From equation (8), if the bias voltage is zero theactuation term with frequency Ω will be zero as well. In order toactuate the first (second) mode for the case of zero DC voltage, thefrequency of the AC voltage should be tuned to half of the first(second) modes natural frequency. Since the AC signal for RFapplications is very small, a DC voltage is used in most of theapplications. Thus, there will be a higher harmonic at two times thenatural frequency of the second mode. An ideal design will avoid any1:2:4 resonance between the first three modes of the system so that thethird mode is not excited from the higher harmonic present in theactuation.

FIG. 3 shows the first four natural frequencies,${{fi} = \frac{\omega_{i}}{2\pi}},$of the T-beam structure as a function of the length ratio (L₃/L₁) or(v₃/v₁). The T-beam structure parameters are: no rigid masses(R_(c)=γ_(c)=R_(t)=γ_(t=)0), equal mass per unit lengths (r₁=r₂=r₃),equal flexural rigidities (α₁=α₂=α₃), equal lengths of the horizontalbeams (v₁=v₂). The solid line denotes analytically computed frequenciesand the symbol “*” denotes frequencies computed using FEM ANSYS model ofthe structure. FIG. 4 shows the first two modes of the T-beam structurewhen the length ratio (L₃/L₁) or (v₃/v₁)=1.3266.

We can use the external detunings to relate the two natural frequenciesup to O(ε) as follows:ω₂=2ω₁(1+εσ_(I))  (24)whereσ_(I)=σ₁−σ₂.  (25)

Here, σ₁ is the internal mistuning between the first two naturalfrequencies from exact 1:2 resonance. The internal resonance between thetwo modes result in modal interaction between the first and second modeswhen either of the modes are directly excited by electrostaticactuation. Thus, there will be a nonzero response of both the first andsecond modes in steady state when either of the modes are exciteddirectly. In comparison, the responses of other modes (which are notcoupled by internal resonances) can be decaying due to damping in thestructure. Thus, the displacements of the beam are approximated usingthe first two internally resonant modes, as follows:v _(i)=ε(A ₁φ_(1i) +A ₂φ_(2i))  (26)u _(i)=ε²(A ₁ ²η_(11i) +A ₂ ²η_(22i)+2A ₁ A ₂η_(12i))  (27)where A_(i) are functions of time, φ_(1i) and φ_(2i) are the modalresponses of the ith beam in first and second modes and are functions ofthe spatial coordinate s _(i). The variables η_(jki) are also functionsof the spatial coordinate s _(i). The eigenfunctions are determined byobtaining V_(i) using equation (18) and the associated characteristicmatrix for the linear system. The axial displacements are of order O(ε²)and are assumed to be caused by transverse displacements. Thisparticular form of the axial displacement is motivated by the form ofaxial displacements in cantilever and clamped-clamped beam problems. Thespatial form of the axial displacement of the ith beam is captured byη_(jki).

The spatial function for the vertical beam (beam 30) axial displacement,η_(jk3), is obtained by writing Lagrangian using the expressions inequation, retaining quadratic nonlinearities (terms up to O(ε³)), andthen requiring L to be stationary with respect to the Lagrangemultiplier λ₁. The spatial functions for beams 26 and 28 axialdisplacements, η_(jk1) and η_(jk2), are determined by including cubicnonlinearities (terms up to O(ε⁴)) in the Lagrangian and then neglectinginertia of the axial displacements. This process of neglecting inertiadue to axial displacement is similar to the one used in the process ofobtaining axial displacements in terms of, transverse displacement inclamped-clamped beam problem. Because the dynamics of the 1:2 internallyresonant beam structure to the first order can be captured by retainingquadratic nonlinearities, the details of finding spatial functions arenot provided here. The spatial functions obtained using the aboveapproach are given by: $\begin{matrix}\left. \begin{matrix}{\eta_{{jk}\quad 1} = {- {\frac{1}{2v_{1}}\left\lbrack {{\int_{0}^{{\overset{\_}{s}}_{1}}{\left( \frac{\partial\phi_{j\quad 1}}{\partial{\overset{\_}{s}}_{1}} \right)\left( \frac{\partial\phi_{k\quad 1}}{\partial{\overset{\_}{s}}_{1}} \right)\quad{\mathbb{d}{\overset{\_}{s}}_{1}}}} -} \right.}}} \\\left. {\frac{v_{\quad 1}}{v_{1} + v_{2}}\left( {\sum\limits_{i = 1}^{2}{\int_{0}^{1}{\frac{v_{1}}{v_{i}}\quad\left( \frac{\partial\phi_{j\quad i}}{\partial{\overset{\_}{s}}_{i}} \right)\left( \frac{\partial\phi_{k\quad i}}{\partial{\overset{\_}{s}}_{i}} \right){\mathbb{d}{\overset{\_}{s}}_{i}}}}} \right){\overset{\_}{s}}_{1}} \right\rbrack\end{matrix} \right\} & (28) \\\left. \begin{matrix}{\eta_{{jk}\quad 2} = {- {\frac{1}{2v_{1}}\left\lbrack {{\frac{v_{1}}{v_{2}}{\int_{0}^{{\overset{\_}{s}}_{2}}{\left( \frac{\partial\phi_{j\quad 2}}{\partial{\overset{\_}{s}}_{2}} \right)\left( \frac{\partial\phi_{k\quad 2}}{\partial{\overset{\_}{s}}_{2}} \right)\quad{\mathbb{d}{\overset{\_}{s}}_{2}}}}} -} \right.}}} \\\left. {\frac{v_{\quad 2}}{v_{1} + v_{2}}\left( {\sum\limits_{i = 1}^{2}{\int_{0}^{1}{\frac{v_{1}}{v_{i}}\quad\left( \frac{\partial\phi_{j\quad i}}{\partial{\overset{\_}{s}}_{i}} \right)\left( \frac{\partial\phi_{k\quad i}}{\partial{\overset{\_}{s}}_{i}} \right){\mathbb{d}{\overset{\_}{s}}_{i}}}}} \right){\overset{\_}{s}}_{2}} \right\rbrack\end{matrix} \right\} & (29) \\{\eta_{{jk}\quad 3} = {{- \frac{1}{2v_{3}}}{\int_{0}^{{\overset{\_}{s}}_{3}}{\left( \frac{\partial\phi_{j\quad 3}}{\partial{\overset{\_}{s}}_{3}} \right)\left( \frac{\partial\phi_{k\quad 3}}{\partial{\overset{\_}{s}}_{3}} \right){\mathbb{d}{\overset{\_}{s}}_{3}}}}}} & (30)\end{matrix}$where (j,k) can take values (1,1), (2,2) or (1,2).

Small changes in system parameters (like lengths of the beam segments,additional mass) can also mistune the system from perfect 1:2 internalresonance. To model this, we introduce mistunings in lengths fromcritical length ratios, and masses for 1:2 internal resonance. Themistunings are defined as follows: $\begin{matrix}\left. \begin{matrix}{{\frac{v_{3}}{v_{1}} = {\left( \frac{v_{3}}{v_{1}} \right)_{c}\left( {1 + {ɛ\sigma}_{L}} \right)}},} & {{R_{t} = {\left( R_{t} \right)_{c} + {ɛ{\hat{R}}_{t}}}},} \\{{R_{c} = {\left( R_{c} \right)_{c} + {ɛ{\hat{R}}_{c}}}},} & {{\gamma_{t} = {\left( \gamma_{t} \right)_{c} + {ɛ{\hat{\gamma}}_{t}}}},} \\{\gamma_{c} = {\left( \gamma_{c} \right)_{c} + {ɛ{{\hat{\gamma}}_{c}.}}}} & \quad\end{matrix} \right\} & (31)\end{matrix}$The parameters denoted as ( )_(c), represent design parameters for whichthe structure exhibits perfect 1:2 internal resonance. σ_(L) representsmistuning in the length ratio v₃/v₁, {circumflex over (R)}_(t) and{circumflex over (R)}_(c) represent mistunings in the tip mass andcentral mass respectively and j, and AC represent mistunings in therotary inertias of the tip mass and central mass, respectively.

The weak nonlinear response of the structure is obtained by averagingthe Lagangian over the time period Tp=4π/Ω of the primary oscillation orthe fast time scale. The evolution of modal amplitudes and phases overslow time scale are determined using the averaged Lagrangian method. Toexplicitly introduce the slow time variables, the time dependent ithmodal amplitude, A_(i), is assumed to be of the following form:$\begin{matrix}{A_{i} = {{p_{i}{\cos\left( {\frac{1}{R_{1}}i\quad{\Omega\tau}} \right)}} + {q_{i}{\sin\left( {\frac{1}{R_{1}}i\quad{\Omega\tau}} \right)}}}} & (32)\end{matrix}$where the p_(i) and q_(i) are quantities dependent on slow time scaleT₁=ετ. The derivative of A_(i) is as follows: $\begin{matrix}\left. \begin{matrix}{{\overset{.}{A}}_{i} = {{\frac{1}{2}i\quad{\Omega\left( {{{- p_{i}}{\sin\left( {\frac{1}{2}i\quad{\Omega\tau}} \right)}} + {q_{i}{\cos\left( {\frac{1}{2}i\quad{\Omega\tau}} \right)}}} \right)}} +}} \\{ɛ\left( {{p_{i}^{\prime}{\cos\left( {\frac{1}{2}i\quad{\Omega\tau}} \right)}} + {q_{i}^{\prime}{\sin\left( {\frac{1}{2}i\quad{\Omega\tau}} \right)}}} \right)}\end{matrix} \right\} & (33)\end{matrix}$where a prime denotes derivative with respect to the slow time scaleT_(i). Recall that a dot denotes derivative with respect to the fasttime scale τ.

Using the above assumptions, and substituting the mistunings inequations (31), (22), and (23), displacements in equations (26) and(27), modal amplitudes in equation (32), and nondimensional parametersintroduced in equations (7), (8), and (16) into the augmentedLagrangian, equation (5), and retaining terms up to O(ε³) results in thefollowing: $\begin{matrix}{L_{aug} = {\left( \frac{EI}{L} \right){\left( {\mathcal{L} + {O\left( ɛ^{4} \right)}} \right).}}} & (34)\end{matrix}$

The Lagrangian L depends only on non-dimensional parameters. Because thetransverse displacements are measured from the static equilibrium, theelectrostatic term with F₀, equation (8), will not appear in theLagrangian. Also, given the way we have defined the axial displacementof beam 30, equation (30), the term with Lagrange multiplier λ₁ is zero.The averaged Lagrangian over the period Tp=4π/Ω is given by:$\begin{matrix}\left. \begin{matrix}{\left\langle \mathcal{L} \right\rangle = {\frac{1}{T_{p}}{\int_{0}^{T_{p}}{\mathcal{L}{\mathbb{d}\tau}}}}} \\{= {ɛ^{2}\left\lbrack {{\frac{r_{1}v_{1}}{2}\left( {\sum\limits_{j = 1}^{2}\Gamma_{j}} \middle| {}_{e = 0}{\frac{1}{2}\left( \frac{j\Omega}{R_{1}} \right)^{2}\left( {p_{j}^{2} + q_{j}^{2}} \right)} \right)} -} \right.}} \\{{\frac{\alpha_{1}}{2v_{1}^{3}}\left( {\sum\limits_{j = 1}^{2}\Gamma_{2 + j}} \middle| {}_{ɛ = 0}{\frac{1}{2}\left( {p_{j}^{2} + q_{j}^{2}} \right)} \right)} +} \\{\left. {\left( {{\overset{\_}{l}}_{1} + {\overset{\_}{l}}_{2}} \right)\frac{{\hat{F}}_{0}}{2}} \right\rbrack +} \\{ɛ^{3}{\frac{r_{1}v_{1}}{2}\left\lbrack {\sum\limits_{j = 1}^{2}\frac{\partial\Gamma_{j}}{\partial ɛ}} \middle| {}_{ɛ = 0}{{\frac{1}{2}\left( \frac{j\Omega}{R_{1}} \right)^{2}\left( {p_{j}^{2} + q_{j}^{2}} \right)} -} \right.}} \\{\left( {\sum\limits_{j = 1}^{2}\Gamma_{j}} \middle| {}_{ɛ = 0}{\frac{j\Omega}{R_{1}}\left( {{p_{j}q_{j}^{\prime}} - {q_{j}p_{j}^{\prime}}} \right)} \right) -} \\{2\left( {N_{1} + {\left( R_{t} \right)_{c}N_{2}}} \right)\left( \frac{\Omega}{R_{1}} \right)^{2}\left( {{p_{1}q_{1}q_{2}} +} \right.} \\{\left. {\frac{1}{2}\left( {p_{1}^{2} - q_{1}^{2}} \right)p_{2}} \right) -} \\{{\frac{\alpha_{1}}{r_{1}v_{1}^{4}}\left( {\sum\limits_{j = 1}^{2}\frac{\partial\Gamma_{2 + j}}{\partial ɛ}} \middle| {}_{ɛ = 0}{\frac{1}{2}\left( {p_{j}^{2} + q_{j}^{2}} \right)} \right)} -} \\{\frac{1}{2}\frac{{\hat{F}}_{1}}{g\quad r_{1}}\left\{ {{\left( {{\Gamma_{f\quad 1}p_{1}\Delta_{1}} + {\Gamma_{f\quad 2}p_{2}\Delta_{2}}} \right)M_{2}} +} \right.} \\\left. \left. {\left( {{\Gamma_{f\quad 3}p_{1}\Delta_{1}} + {\Gamma_{f\quad 4}p_{2}\Delta_{2}}} \right)M_{1}} \right\} \right\rbrack\end{matrix} \right\} & (35)\end{matrix}$where Γ_(j), N₁, N₂ and Γ_(fj)(j=1, 2, 3, 4) are defined in Appendix.Δ₁=1 when the first mode is in the primary resonance and zero when thesecond mode is in the primary resonance. Similarly, Δ₂=1 when the secondmode is excited and zero when the first mode is excited. Although weonly explicitly account for mistunings related to the parameters inequation (31), the Lagrangian formulated above can also account formistunings in the variations in mass per unit lengths r_(i), flexuralrigidities α_(i), and bottom beam length ratio v₂/v₁. These variationswill also result in mistuning the first two modes of the structure awayfrom 1:2 internal resonance. The terms up to O(ε²) represent linearterms and so the stiffness and inertia terms are related as follows:$\quad\begin{matrix}\left. \begin{matrix}{{\left. {\frac{\alpha_{1}}{r_{1}v_{1}^{4}}\Gamma_{3}} \right|_{ɛ = 0} = \left. {\omega_{1}^{2}\Gamma_{1}} \right|_{ɛ = 0}},} \\{\left. {\frac{\alpha_{1}}{r_{1}v_{1}^{4}}\Gamma_{4}} \right|_{ɛ = 0} = \left. {\omega_{2}^{2}\Gamma_{2}} \middle| {}_{ɛ = 0}. \right.}\end{matrix} \right\} & (36)\end{matrix}$We substitute equation (36) and equations (22) and (23), in to equation(35) and then use the extended Hamilton's principle to deriveEuler-Lagrange equations of motion. The resulting equations of motion,including the effect of the scaled modal damping ({circumflex over(ξ)}_(i)) for ith modes are: $\begin{matrix}{\left. \begin{matrix}{{p_{1}^{\prime} = {{{- {\overset{\_}{\sigma}}_{1}}\omega_{1}q_{1}} - {{\hat{\zeta}}_{1}\omega_{1}p_{1}} + {\omega_{1}{\Lambda_{1}\left( {{p_{1}q_{2}} - {q_{1}p_{2}}} \right)}}}},} \\{{q_{1}^{\prime} = {{{\overset{\_}{\sigma}}_{1}\omega_{1}p_{1}} - {{\hat{\zeta}}_{1}\omega_{1}q_{1}} - {\omega_{1}{\Lambda_{1}\left( {{p_{1}p_{2}} + {q_{1}q_{2}}} \right)}} - E_{1}}},} \\{{p_{2}^{\prime} = {{{- {\overset{\_}{\sigma}}_{2}}\omega_{2}q_{2}} - {{\hat{\zeta}}_{2}\omega_{2}p_{2}} + {2\omega_{1}\Lambda_{2}p_{1}q_{1}}}},} \\{{q_{2}^{\prime} = {{{\overset{\_}{\sigma}}_{2}\omega_{2}p_{2}} - {{\hat{\zeta}}_{2}\omega_{2}q_{2}} - {\omega_{1}{\Lambda_{2}\left( {p_{1}^{2} - q_{1}^{2}} \right)}} - E_{2}}},}\end{matrix} \right\}{where}} & (37) \\\left. {\begin{matrix}{{\overset{\_}{\sigma}}_{j} = {\sigma_{j} + \frac{1}{\left. {2\omega_{j}^{2}\Gamma_{j}} \right|_{ɛ = 0}}}} & \quad \\{\begin{bmatrix}{{\left( {{\omega_{j}^{2}S_{1j}} - {\frac{\alpha_{1}}{r_{1}v_{1}^{4}}S_{2j}}} \right)\sigma_{L}} + \omega_{j}^{2}} \\\begin{pmatrix}{{S_{3j}{\hat{R}}_{t}} + {S_{4j}{\hat{R}}_{c}} +} \\{{S_{5j}{\hat{\gamma}}_{t}} + {S_{6j}{\hat{\gamma}}_{c}}}\end{pmatrix}\end{bmatrix},} & \quad \\{{E_{1} = {\frac{1}{4}\frac{{\hat{F}}_{1}}{\left. {g\quad r_{1}\omega_{1}\Gamma_{1}} \right|_{ɛ = 0}}\left( {{\Gamma_{f\quad 1}M_{2}} + {\Gamma_{f\quad 3}M_{1}}} \right)\Delta_{1}}},} & \quad \\{{E_{2} = {\frac{1}{4}\frac{{\hat{F}}_{1}}{\left. {g\quad r_{1}\omega_{2}\Gamma_{2}} \right|_{ɛ = 0}}\left( {{\Gamma_{f\quad 2}M_{2}} + {\Gamma_{f\quad 4}M_{1}}} \right)\Delta_{2}}},} & \quad \\{{\Lambda_{1} = \frac{\left( {N_{1} + {\left( R_{t} \right)_{c}\left( N_{2} \right)}} \right.}{\left. \Gamma_{1} \right|_{ɛ = 0}}},} & {{\Lambda_{2} = \frac{\left( {N_{1} + {\left( R_{t} \right)_{c}N_{2}}} \right)}{\left. {4\Gamma_{2}} \right|_{ɛ = 0}}},} \\{{S_{1\quad j} = \left. \frac{\partial^{2}\Gamma_{j}}{{\partial ɛ}{\partial\sigma_{L}}} \right|_{ɛ = 0}},} & {{S_{2\quad j} = \left. \frac{\partial^{2}\Gamma_{2 + j}}{{\partial ɛ}{\partial\sigma_{L}}} \right|_{ɛ = 0}},} \\{{S_{3\quad j} = \left. \frac{\partial^{2}\Gamma_{j}}{{\partial ɛ}{\partial R_{t}}} \right|_{ɛ = 0}},} & {{S_{4\quad j} = \left. \frac{\partial^{2}\Gamma_{j}}{{\partial ɛ}{\partial R_{c}}} \right|_{ɛ = 0}},} \\{{S_{5\quad j} = \left. \frac{\partial^{2}\Gamma_{j}}{{\partial ɛ}{\partial{\hat{\gamma}}_{t}}} \right|_{ɛ = 0}},} & {S_{6\quad j} = \left. \frac{\partial^{2}\Gamma_{j}}{{\partial ɛ}{\partial{\hat{\gamma}}_{c}}} \middle| {}_{ɛ = 0}. \right.}\end{matrix}\quad{\quad\quad}} \right\} & (38)\end{matrix}$The actual modal damping, ξ_(i), of the ith mode is related to thescaled damping, ({circumflex over (ξ)}_(i)), as follows:ζ_(t)=ε{circumflex over (ζ)}_(i).  (39)The {circumflex over (σ)}_(i) in averaged equations (37) represents theeffective external mistuning of the ith mode. This effective mistuningincludes the sensitivities S_(ji) of different structure parameters asgiven in the equation (38). An effective internal mistuning of the twomodes from 1:2 internal resonance, σ _(i), can thus be written asfollows:σ _(I)= σ ₁− σ ₂.  (40)

The averaged equations (37) obtained here are identical to the averagedequations obtained for quadratically coupled internally resonantoscillators when excited externally in the first or second mode. Thesequadratically coupled oscillators are studied by many researchers forequilibrium solutions and bifurcations re suiting in complex dynamics.

Response of an Illustrative Structure

Consider the T-shaped structure discussed while illustrating the resultsof linear analysis. The specific nominal T-beam structure had no rigidmasses, and all the three beam segments have the same mass per unitlengths and the same flexural rigidities for all the beams. Further, thelengths and widths of the two bottom beams are equal. We choose nominalparameters as the parameters of the bottom left beam, the beam 26. Theseassumptions are written in terms of parameter values defined in theequations (7) and (16), as follows: $\begin{matrix}\left. \begin{matrix}{{r_{j} = 1},} & {{\alpha_{j} = 1},} & {{v_{1} = 1},} & {{v_{2} = 1},} \\{{\left( R_{c} \right)_{c} = 0},} & {{\left( R_{t} \right)_{c} = 0},} & {{\left( \gamma_{c} \right)_{c} = 0},} & {{\left( \gamma_{t} \right)_{c} = 0},} \\{{\left. v_{3} \right|_{c} = 1.3266},} & \quad & \quad & \quad\end{matrix} \right\} & (41)\end{matrix}$where j=1, 2, 3. The non-dimensional natural frequencies of the systemobtained by linear analysis are as follows:ω₁=1.699, ω₂=3.398.  (42)The first two mode shapes are computed analytically and the mode shapesare normalized to have Γ_(1|E=0)=Γ_(2|ε=0)=1.

The microresonator structure's actual dimension are assumed to be thesame as specified in equation (20). The gap between the horizontal beams26, 28 and the second electrode 20 is fixed at d=1 μm. The gap betweenthe vertical beam 30 and the first electrode 24 is also fixed at d₃=1μm. All these dimensions are chosen keeping in mind the fabricationconstraints. Using the permittivity of air and the mode shapes, thedifferent terms defining forcing term reduces to just a function ofapplied voltage, as given below: $\begin{matrix}\left. \begin{matrix}{{\left( {\frac{1}{4}\frac{{\hat{F}}_{1}}{g\quad r_{1}}} \right) = {\frac{2.83{TV}_{b}V}{ɛ^{3}} \times 10^{- 6}}},} \\{{\Gamma_{f\quad 1} = {\Gamma_{f\quad 4} = 0}},} \\{{\Gamma_{f\quad 2} = 0.421},} \\{\Gamma_{f\quad 3} = 0.868}\end{matrix} \right\} & (43)\end{matrix}$

The parameters in the above equation (43) suggest that, to the firstorder approximation, the first mode cannot be excited directly by usingthe second electrode 20, and similarly the second mode cannot bedirectly excited by the first electrode 24. This filtering of the firstmode frequency when actuating by the second electrode 20 (and secondmode frequency when actuating by the first electrode 24) is due to themode shape of the system. This filtering characteristic will also bevalid in general for microresonator structures as long as the symmetryof the structure is maintained.

The other parameters required to compute all the coefficients in theaveraged equations (37) for this system are as follows: $\begin{matrix}\left. \begin{matrix}{{S_{11} = 0.998},} & {{S_{21} = 33.989},} & {{S_{31} = 3.714},} \\{{S_{41} = 0},} & {{S_{51} = 2.602},} & {{S_{61} = 0.101},} \\{{S_{12} = 0.6357},} & {{S_{22} = 0},} & {{S_{32} = 0.636},} \\{{S_{42} = 0.958},} & {{S_{52} = 0},} & {S_{62} = 0} \\{{\Lambda_{1} = 0.7634},} & {\Lambda_{2} = {0.1908.}} & \quad\end{matrix} \right\} & (44)\end{matrix}$The parameters S_(ij) determine the sensitivity of jth natural frequencyto changes in different parameters, see Equation (38). The values of S₁₁and S₂₁ suggest that the first mode is very sensitive to any changes inthe length ratio (v₃/v₁). However, attaching a central rigid mass,parameter S₄₁, does not affect the natural frequency of the first mode.The parameters S₅₂ and S₆₂ suggest that the rotational inertia of therigid masses do not affect the second mode natural frequency. Λ₁ and Λ₂represent the strength of nonlinear coupling between the two modes.

The quality factor Q (as traditionally defined for liner systems) forthe response of a microresonator in ith mode is (1/2ζ_(i)), where (ζi isthe modal damping for ith mode. Since, quality factor is an importantperformance parameter for a microresonator, we compute the response ofthis structure for different values of damping coefficients. Thestructure is assumed to have perfect internal resonance, σ ₁=0, and theeffects of different parameter sensitivities on the response are notdiscussed here. For a given excitation voltage and excitation in eithermode, the decrease in damping of the first and second mode can result inHopf bifurcations and thereby period doubling bifurcations and chaoticmotions of the beam. The internal mistuning of the first two modes from1:2 internal resonance also plays a critical role in determining theexistence of Hopf bifurcation in the system dynamics. If the two modesare in perfect internal resonance, the system will not have any Hopfbifurcation when the second mode is excited directly. However, when thefirst mode is excited directly (Δ₁=1); the system with the two modes inperfect internal resonance can still undergo Hopf bifurcation.

The response of the structure is simulated in bifurcation andcontinuation software AUTO with scaling parameter ε=0.01 to scale theresponse to O(l). First, we consider a structure with low quality factorQ=500 for both the modes, and thus fix the value of damping parameters,ζ₁ and ζ₂, to 0.001. The scaled modal damping is then obtained usingequation (39). The response obtained using AUTO is then scaled back tothe actual beam displacements using equations (7), (16), and (27). Whenthe structure is excited directly in the first mode, the structure isactuated using the first electrode 24. For the direct excitation of thesecond mode, the second electrode 20 is used for actuation. FIG. 5 showsthe response for the primary resonance of the first mode in terms of thedisplacements at the tip of the upper beam for first mode (FIG. 5(a))and at the junction of the three beams for second mode (FIG. 5(b)) fordifferent input voltages. The modal response thus plotted is the maximumdisplacement of the structure in that particular mode. The responseundergoes Hopf-bifurcation for higher voltages. When the first mode isdirectly excited, the response in both the modes remains non-zero overthe entire bandwidth of interest. The sensor output signal at the secondelectrode 20 is at double the input frequency and also act as a mixer(upconversion) and filter for the input signal.

Similarly, FIG. 6 shows the response for the primary resonance of secondmode. While FIG. 6(a) shows the transverse (horizontal) displacement ofthe tip of the upper beam 30, FIG. 6(b) shows the response of thejunction of the three beams 26, 28 and 30. Interestingly, when thesecond mode is excited, the response of the first mode, and hence thetip of the upper beam 30, is non-zero only for a certain range offrequency, and thus the response in first mode per forms a filteringaction on the input signal. Further, in this case the first electrode 24output signal is at half the input frequency and thus the resonator actsas a mixer (downconversion).

FIG. 7 depict the horizontal displacement of the tip of the upper beam30. FIG. 7(a) shows the response in the first mode when the first modeis excited directly for a structure having a higher quality factorQ=5000 (achievable easily for micro resonators in vacuum, may bedifficult in air). The response again undergoes Hopf bifurcation forhigher voltages. The appearance of Hopf bifurcation depends on thestrength of excitation in comparison to the structure damping. FIG. 7(b)shows the response in the first mode for the same structure when thesecond mode is in primary resonance, that is the structure is excited bythe electrode 20. Since the two modes of the microresonator structureare assumed to be in perfect internal resonance, there is no Hopfbifurcation in the response of the system in FIG. 7(b). Further, theresponse changes appreciably with voltage when either of the modes areexcited. This may limit the power handling capacity of such resonators.

The Γ_(j) and other undefined variables in Lagrangian are defined below.These variables are constant with respect to time and are determined bythe mode shapes of the structure. $\begin{matrix}\left. \begin{matrix}{\Gamma_{j} = {\left( {\sum\limits_{i = 1}^{3}{\int_{0}^{1}{\frac{r_{i}v_{i}}{r_{1}v_{1}}\phi_{ji}^{2}{\mathbb{d}{\overset{\_}{s}}_{i}}}}} \right) + \left( {{\frac{r_{3}v_{3}}{r_{1}v_{1}}\left( {1 + R_{t}} \right)} +} \right.}} \\\left. {\left. {R_{c}\left( {1 + \frac{r_{2}v_{2}}{r_{1}v_{1}}} \right)} \right)\phi_{j\quad 1}^{2}} \middle| {}_{{\overset{\_}{s}}_{1} = 1} + \right. \\\left. {\frac{\gamma_{c}}{r_{1}v_{1}^{3}}\left( \frac{\partial\phi_{j\quad 1}}{\partial{\overset{\_}{s}}_{1}} \right)^{2}} \middle| {}_{{\overset{\_}{s}}_{1} = 1}{{+ R_{t}}\frac{r_{3}v_{3}}{r_{1}v_{1}}\phi_{j\quad 3}^{2}} \middle| {}_{{\overset{\_}{s}}_{3} = 1} + \right. \\\left. {\frac{\gamma_{t}}{r_{1}v_{1}^{3}}\left( \frac{v_{1}}{v_{3}} \right)^{2}\left( \frac{\partial\phi_{j\quad 3}}{\partial{\overset{\_}{s}}_{3}} \right)^{2}} \right|_{{\overset{\_}{s}}_{3} = 1}\end{matrix} \right\} & (45) \\{\Gamma_{2 + j} = {\sum\limits_{i = 1}^{3}{\int_{0}^{1}{\frac{\alpha_{i}v_{1}^{3}}{v_{i}^{3}\alpha_{1}}\left( \frac{\partial^{2}\phi_{ji}}{\partial{\overset{\_}{s}}_{i}^{2}} \right)^{2}{\mathbb{d}{\overset{\_}{s}}_{i}}}}}} & (46)\end{matrix}$where j=1 and 2. The Γ_(j) and Γ_(2+j) are dependent on the parameter εas the terms appearing in the above equations like v₃/v₁, R_(t), R_(c)are dependent on the ε as defined by the equations (31).

The other three terms N₁, N₂ and Γ_(f) are also dependent on the modeshapes. These terms are as follows: $\begin{matrix}{N_{1} = {\int_{0}^{1}{I{\mathbb{d}{\overset{\_}{s}}_{3}}}}} & (47) \\{N_{2} = \left. I \right|_{{\overset{\_}{s}}_{3} = 1}} & (48) \\{\Gamma_{fj} = {{\int_{1 - {\overset{\_}{l}}_{1}}^{1}{\phi_{j\quad 1}{\mathbb{d}{\overset{\_}{s}}_{1}}}} + {\frac{b_{2}v_{2}}{b_{1}v_{1}}\begin{matrix}{{\int_{1 - {\overset{\_}{l}}_{2}}^{1}{\phi_{j\quad 2}{\mathbb{d}{\overset{\_}{s}}_{2}}}},} & {{j = 1},2,}\end{matrix}}}} & (49) \\{\begin{matrix}{{\Gamma_{fk} = \left. \frac{{\mathbb{d}b_{3}}v_{3}}{{\mathbb{d}_{3}b_{1}}v_{1}} \middle| {}_{ɛ = 0}{\int_{1 - {\overset{\_}{l}}_{3}}^{1}{\phi_{k\quad 3}{\mathbb{d}{\overset{\_}{s}}_{3}}}} \right.},} & {{k = 3},4,}\end{matrix}{where}{{j = 1},2,{and}}} & (50) \\\left. \begin{matrix}{I = \left\{ {2\frac{r_{3}}{r_{1}}{\frac{v_{3}}{v_{1}}\left\lbrack {\left( \eta_{111} \middle| {}_{{\overset{\_}{s}}_{1} = 1}{\phi_{23} - \eta_{121}} \middle| {}_{{\overset{\_}{s}}_{1} = 1}\phi_{13} \right) -} \right.}} \right.} \\\left. \left. \left. \left( \phi_{21} \middle| {}_{{\overset{\_}{s}}_{1} = 1}{\eta_{113} - \phi_{11}} \middle| {}_{{\overset{\_}{s}}_{1} = 1}\eta_{123} \right) \right\rbrack \right\} \right|_{ɛ = 0}\end{matrix} \right\} & (51)\end{matrix}$

Referring now to FIG. 8, a second embodiment of the present invention isshown. The micromechanical resonator 110 includes a structure 111 havinga first component 112 and a second component 114 generally defining anL-shaped configuration, where the second component 114 is connected tosubstrate 15. The resonator 110 further includes a first mass 116connected to both components 112, 114, a second mass 118 connected tothe free end of the first component 112, a first electrode 124positioned adjacent to the first component 112, and a second electrodepositioned adjacent to the second component 114. As with the resonatordescribed above, the resonator embodies two modes, a lower naturalfrequency mode (the first mode) and a higher frequency mode (the secondmode). The second mode, which may be induced by the second electrode120, is embodied by both the first and second components 112, 114.Specifically, the second mode includes pivoting movement of thestructure 111 about a point near the base 115. Additionally, the firstmode, which also includes pivoting movement about the substrate 115, isembodied by both the first component 112 and the second component 114.As with the resonator 10 described above, the first mode is induced bythe second mode through non-linear modal interaction. Due to theunsymmetrical nature of the structure 111, the first component 112 willmove or flex parallel to the horizontal axis when the second componentis deflecting or flexing along the vertical direction.

Referring now to FIG. 9, a third embodiment of the present invention isshown. The micromechanical resonator 210 includes a structure 111 havinga second component 214 extending between two points on a base orsubstrate 215 and a first component 214, a third component 250 and afourth component 252, each extending generally vertically from thesecond component such that the structure 111 generally defines ancomb-shaped configuration. The resonator 110 further includes aplurality of first masses 116 connected to intersection points betweenthe second component 214 and the perpendicular components 212, 250, 252.Additionally, the first component 214 includes an electrode 220positioned adjacent thereto and each of the perpendicular components212, 250, 252 includes a pair of electrodes 266 positioned on oppositesides thereof. The resonator 210 embodies a second mode includingmovement generally along the vertical axis and a first mode includingmovement generally along the horizontal axis. As with the resonator 10described above, one of the modes may be induced by the electrode 220 orthe pairs of electrodes 266 and the other mode is induced by non-linearinternal resonance between the two modes. Also similarly to theresonator 10 described above, the first mode is linearly decoupled fromthe second mode when the second component 214 is vibrating at afrequency twice the natural frequency of the first mode.

It is therefore intended that the foregoing detailed description beregarded as illustrative rather than limiting, and that it be understoodthat it is the following claims, including all equivalents, that areintended to define the spirit and scope of this invention.

1. A micromechanical resonator comprising a structure configured todefine a first mode and a second mode and to permit non-linear internalresonance between the first and second modes.
 2. A micromechanicalresonator as in claim 1, the structure configured to permit 1:2non-linear internal resonance between the second mode and the firstmode.
 3. A micromechanical resonator as in claim 1, further comprisingan actuator configured to resonate the structure and induce the secondmode.
 4. A micromechanical resonator as in claim 1, the second modeembodied by at least a second component extending along a seconddirection and the first mode embodied by at least a first componentextending along a first direction.
 5. A micromechanical resonator as inclaim 4, the first direction generally perpendicular to the seconddirection.
 6. A micromechanical resonator as in claim 5, the first andsecond components generally defining a T-shaped configuration.
 7. Amicromechanical resonator as in claim 6, the second component having afirst beam and a second beam each defining a beam length, and the firstcomponent having a first component length equal to approximately 133% ofthe beam length of at least one of the first and second beams.
 8. Amicromechanical resonator as in claim 4, further comprising a masscoupled with at least one of the first and second components.
 9. Amicromechanical resonator as in claim 8, the mass positioned adjacent toan intersection point between the first component and the secondcomponent.
 10. A micromechanical resonator as in claim 9, furthercomprising a second mass positioned adjacent to a second end of thefirst component.
 11. A micromechanical resonator as in claim 4, thefirst component defining at least one first natural resonance frequencyand the second component defining at least one second natural resonancefrequency, the first and second modes substantially completelynon-linearly coupled with each other while the second component vibratesat a frequency approximately twice the at least one first naturalresonance frequency.
 12. A micromechanical resonator as in claim 5, thefirst and second components generally defining an L-shapedconfiguration.
 13. A micromechanical resonator as in claim 5, furthercomprising a third component and a fourth component each extendinggenerally parallel to the first component.
 14. A micromechanicalresonator comprising a structure having a first component embodying afirst mode and a second component embodying a second mode, the firstcomponent defining at least one first natural resonance frequency andthe second component defining at least one second natural resonancefrequency, the first and second modes substantially completelynon-linearly coupled with each other while the second component vibratesat a frequency approximately twice the at least one first naturalresonance frequency.
 15. A micromechanical resonator as in claim 14, thefirst and second components generally defining a T-shaped configuration.16. A micromechanical resonator as in claim 14, further comprising athird component and a fourth component each extending generally parallelto the first component.
 17. A micromechanical resonator as in claim 14,the first frequency is approximately one half of the second frequency.18. A micromechanical resonator comprising: a structure having a firstcomponent and a second component; and an actuator configured to induceresonant excitation of the second component at a second frequency;wherein the second component is positioned with respect to the firstcomponent such that the resonant excitation of the second component atthe second frequency induces resonant excitation of the first componentat a first frequency.
 19. A micromechanical resonator as in claim 15,the first frequency is approximately one half of the second frequency.20. A micromechanical resonator as in claim 18, the first and secondcomponents generally defining a T-shaped configuration.
 21. Amicromechanical resonator as in claim 18, the first and secondcomponents generally defining an L-shaped configuration.
 22. Amicromechanical resonator as in claim 18, further comprising a thirdcomponent and a fourth component each extending generally parallel tothe first component.